black
Historically,
[1]} first introduced the notion of confidence sequence, and the idea was further studied by [2]} and [3]}.
blackA recent line of work such as [4]}, [5]}, [6]}, [7]}, [8]} has brought this idea back to researcher' attention.

Note that the concept of confidence sequences has a close connection to “safe testing” [9]}.
The idea of gambling for confidence sequences (or equivalently, time-uniform concentration inequalities) can be found in several different areas.
Shafer and Vovk have advocated to use the idea of gambling to establish a game-theoretic theory of probability [10]}, [11]}. Building upon the gambling approach,
[12]} proposed to test by betting.
[13]}
first pioneered the idea of using betting algorithms and numerically inverting the wealth processes into confidence intervals.
[7]} explored a similar idea, proposed to use the notion of regret, and constructed a time-uniform concentration inequality that achieves the law of iterated logarithm for sub-Gaussian random vectors in Banach spaces.
[5]} further investigated the idea of gambling for confidence sequences, and analyzed several heuristic betting algorithms.
[8]} recently examined the idea of [17]}'s universal portfolio and its regret analysis to construct tight confidence sequences based on a numerical method.

For the detailed literature survey on the idea of gambling for confidence sequences, we refer an interested reader to [5]} and [8]}, and references therein.
